No, that one is based on the empirical reality that there is significant amounts of nesting hierarchical structure in the sequences of the clade of P-loop NTPases that make up the catalytic hexamer proteins.
I believe I linked you a phylogenetic tree of related proteins here.
You see Gilbert, here’s the problem. You already accept this type of evidence, explicitly. In your advancing Gpuccio’s FI calculations, you are accepting phylogenetic inferences of relatedness.
So which one is it going to be, you accept or reject statistically significant nesting hierarchical structure in sequences of of shared similar genes as evidence for relatedness, or you do not?